I currently have a 5c which last night I backed up with iCloud in its settings. Today I'm getting a 6s and I want to transfer contacts, photos , iTunes music etc to. Should I take out my current sim and put it in my new phone or can I just log into my iCloud on the new phone and everything will be there? What is the easiest way?
Assuming you have a paid iCloud subscription that allowed you enough storage space to backup everything, yes, you can just login to your iCloud on the 6s and restore the backup. If you do not have a paid iCloud account, chances are you don't have enough space (free plan is 5GB) and you will need to do a local backup to a computer using iTunes, then restore also using iTunes.
Contacts are not typically stored on the SIM anymore. It's more so just for cell provider activation and service.
If you want to transfer from one iPhone to another, I always do the full backup to the computer (not iCloud) and then restore on the new iPhone. iCloud backups do not backup everything.
Using your iCloud login, your Contacts, Calendars, etc. should already by sync'd between your iPhone and the iCloud servers. When you turn on the new iPhone and enter your AppleID/password, those things will automatically be sync'd even before/without you do the restore.
Nothing is stored on the SIM, so whether you use your old SIM or a new one won't affect much on the phone. To transfer settings and so on to your new phone, you do need to restore from a backup. The iCloud backup should set up your new phone mostly as your old one is.
It's true that if you only use free iCloud with 5 Gb of space, an iCloud backup will be missing some things compared to an iTunes backup. iCloud backs up a list of your apps, and their positions on the home screen, but not the apps themselves or any data associated with them. So, when restoring from an iCloud backup, the phone will re-download all your apps over your Wi-Fi network, so they will come back, but you'll lose your progress in any games, for example.